"I just want to apologise to the fans. They're the only people with the badge on today who can hold their head high, because we can't."

Speaking to Match of the Day, Solskjaer added: "We were beaten on all aspects today and the only place we beat Everton - I can't say we beat them - but we had a fantastic support and I just want to apologise for the performance we turned out."

Goals from Richarlison, Gylfi Sigurdsson, Lucas Digne and Theo Walcott handed United their biggest defeat of the season in a game where the visitors did not register their first shot on target until the 86th minute.

United are next in action against local rivals Manchester City on Wednesday.

"We just didn't perform. That was not worthy of a Manchester United team," said the Norwegian, who was appointed permanent manager in March.

"That performance is not good enough for a Manchester United team, from me to players, we let the fans down, we let the club down. That performance is difficult to describe because it is so bad.

"They beat us on all the basics. We were beaten on all the ingredients you need, added to the talent. There is no place you can hide on the pitch.

"We hold our hands up and apologise to everyone associated with the club. We have a perfect chance to make amends on Wednesday."

The visitors collectively covered 8km less than Everton during the defeat, but Solskjaer said their decision making was a much as to blame as their lack of running.

Asked if his players cared, Solskjaer said: "You have got to ask them. I have asked them. Of course if you want to play at this club, it has to be more.

"I want my team to be the hardest working team in the league, that is what we were under Sir Alex [Ferguson] - [Ryan] Giggs, Becks [David Beckham], Gary Neville, Denis [Irwin] - no matter how much talent you have got, you run more than anyone.

"You can't change your whole squad. One step at a time. I am going to be successful here and there are players who won't be part of that."

Meanwhile, Manchester United captain David de Gea said it was "difficult to put into words" what he was feeling after the defeat.

"As captain, I want to say that the performance of the team today, on top of the result, was not up to the standard expected when you pull on this shirt," he said on Twitter.

"We know that we have to improve but words mean little: we have to give everything for this club. Thank you for your unconditional support."

The stats - how bad were United?

Manchester United did not manage a shot on target against Everton until the 86th minute

Where do you start? United only managed one shot on target against Everton, have been out-run by their opponents in 15 of 17 games in the Premier League under Solskjaer and have now conceded more goals than they have in a league season for 40 years.

Manchester United have gone 11 consecutive matches without a clean sheet in all competitions for the first time since December 1998.

They have conceded 48 Premier League goals this campaign, their most in a single season in the competition and more than any league campaign since 1978-79, when they let in 63.

This was Everton's biggest win over Manchester United in any competition since a 5-0 victory in October 1984, and the Reds' biggest loss in the Premier League since a 4-0 defeat by Chelsea in October 2016.

It was a fifth consecutive away defeat across all competitions for United, their longest such run since March 1981, when they lost five in a row under Dave Sexton.

They have also lost three successive Premier League away games for the first time since a run between December 1995 and January 1996.

United were outrun by Everton by 8.03km, their highest deficit since they lost to 3-0 to Tottenham in August at Old Trafford, when they covered 10.2km fewer than Spurs.

Analysis - 'Solskjaer must find a way to reboot this side'

Phil McNulty, BBC Sport chief football writer:

Manchester United manager Ole Gunnar Solskjaer was in full-on apology mode after they walked into a severe mauling against Everton at Goodison Park.

And once the apologies are accepted, Solskjaer must find a way to reboot this United side after a performance lacking in guts, heart and quality that will rank alongside the worst they have given in recent memory.

Paul Pogba may enjoy the attention of being a big-time player but it is time he actually performed like one. United were overrun in every area of the pitch. Romelu Lukaku looked lost; United were taken apart by Marco Silva's side.

Solskjaer will, of course, be given time to attack this task but the bottom line is that this Manchester United side have deficiencies in all areas and even the normally reliable goalkeeper David de Gea cannot escape criticism.

He was the one Manchester United could rely on but not in this inconsistent campaign - on this evidence, there is currently nothing Solskjaer can bank on. The Norwegian faces a huge rebuilding programme.

What the pundits said - 'players' heads are on the line'

Former Manchester United defender Gary Neville, on Sky Sports: "I'm furious, the fact he's had to go over and apologise for them. I've often said if you've got weeds in the garden, you have got to get rid of them.

"But there is some Japanese knotweed at that football club and it's attacking the foundations of the house, it needs dealing with properly. The fans aren't going to turn on Ole Gunnar Solskjaer; the players' heads are on the line.

"Manchester United players, honestly, it's embarrassing just watching some of those out there. I am absolutely furious. Full belief in Ole to do what he needs to to do, the club obviously will support him. But there is a big problem and I have to say that performance today was rancid."

Former Manchester United midfielder Paul Ince on BBC Radio 5 Live: "The way Manchester United are performing right now, you have to ask if Ole Gunnar Solskjaer is the right man? You cannot be getting embarrassed at Goodison Park like this."

Premier League-winning defender Robert Huth on 5 Live: "Manchester United could have easily waited until the end of the season and assessed everything about Ole Gunnar Solskjaer before making a decision.

"They panicked and took a big decision. Who knows if he will be here next season - football is crazy sometimes."

Ex-Bolton manager Owen Coyle on 5 Live: "Manchester United need investment and players. Every signing is a gamble but you have to get as many right as possible to challenge for the Premier League."

Former Blackburn striker Chris Sutton on 5 Live: "I still think Ole Gunnar Solskjaer is the right appointment. Jose Mourinho made a mess and Ole has to pick up the pieces."

What you said...

Aaron Harte Hughes: I'd prefer if United play the reserves against City, more fight in them. Don't know how United are going to attract big stars this summer when all they have to look forward to is playing in the Europa League.

John Rambo Nsona: Remember when Mourinho said one of his greatest achievements in football was finishing second with this Man Utd squad and we all laughed? Yeah, that.

Sam Baines: Much more wrong with this club than just the starting line up. Thought Solskjaer was a good appointment. Starting to have second thoughts now. This is embarrassing.

Gav Carlton: Got to question the players' attitudes; most realistic fans know a good majority of these players aren't good enough.

Ryan Young: United are finished. Why did they appoint a manager after three months! He's clearly out of his depth. The players are a shambles; all of them trying to get new contracts they don't deserve. Sell the lot!